Exponential State Observer Design for a Class of Uncertain Chaotic and Non-Chaotic Systems

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Exponential State Observer DesignUncertain Chaotic

Professor, Department of Electrical Engineering,

ABSTRACT In this paper, a class of uncertain chaotic and nonchaotic systems is firstly introduced and the state observation problem of such systems is explored. Based on the time-domain approach with integradifferential equalities, an exponential state observer for a class of uncertain nonlinear systems is established to guarantee the global exponential stability of the resulting error system. Besides, the guaranteed exponential convergence rate can be calculated correctly. Finally, numerical simulations are presented to exhibit the feasibility and effectiveness of the obtained results. Key Words: Chaotic system, state observer design, uncertain systems, exponential convergence rate 1. INTRODUCTION In recent years, various chaotic systemswidely explored by scholars; see, for example, [1and the references therein. Frequently, chaotic signals are often the main cause of system instability and violent oscillations. Moreover, chaos often occurs in various engineering systems and applied physicsinstance, ecological systems, secure communicatiand system identification. Based on practical considerations, not all state variables of most systems can be measured. Furthermore, the design of the state estimator is an important work when the sensor fails.Undoubtedly, the state observer design of with both chaos and uncertainties tends to be more difficult than those without chaos and uncertainties.For the foregoing reasons, the observer design of uncertain chaotic systems is really significant essential. In this paper, the observability problem for uncertain nonlinear chaotic or non-chaotic investigated. By using the time-domain approach with

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Exponential State Observer Design for a Class Uncertain Chaotic and Non-Chaotic Systems

Yeong-Jeu Sun f Electrical Engineering, I-Shou University, Kaohsiung, Taiwan

In this paper, a class of uncertain chaotic and non-chaotic systems is firstly introduced and the state observation problem of such systems is explored.

domain approach with integral and differential equalities, an exponential state observer for a class of uncertain nonlinear systems is established to guarantee the global exponential stability of the resulting error system. Besides, the guaranteed exponential convergence rate can be calculated correctly. Finally, numerical simulations are presented to exhibit the feasibility and

Chaotic system, state observer design, uncertain systems, exponential convergence rate

chaotic systems have been ; see, for example, [1-8]

Frequently, chaotic signals are often the main cause of system instability and

Moreover, chaos often occurs in various engineering systems and applied physics; for instance, ecological systems, secure communication,

Based on practical considerations, not all state variables of most systems

Furthermore, the design of the state estimator is an important work when the sensor fails.

design of systems with both chaos and uncertainties tends to be more

without chaos and uncertainties. the foregoing reasons, the observer design of

significant and

lity problem for a class of chaotic systems is

domain approach with

integral and differential equalitiesobserver for a class of uncertain will be provided to ensure the global exponential stability of the resulting error system. In guaranteed exponential convergence rate can be precisely calculated. Finally, some numerical simulations will be given to demonstrate the effectiveness of the main result. This paper is organized as follows. The problem formulation and main results 2. Several numerical simulations are3 to illustrate the main result. Finally, conclusion remarks are drawn in Section paper, nℜ denotes the n-dimensional

xxx T ⋅=: denotes the Euclidean norm of the

vector x, and a denotes the

number a. 2. PROBLEM FORMULATION AND MAI

RESULTS In this paper, we explore the following nonlinear systems:

( ) ( ) ( ) ( )( ),,, 32111 txtxtxftx ∆=& ( ) ( ) ( ) ( )( ),, 31222 txtxftaxtx +=& ( ) ( )( ) ( )( ),14333 txftxftx +=&

( ) ( ) ,0,3 ≥∀= ttbxty

( ) ( ) ( )[ ] [T xxxxxx 2010321 000 = where ( ) ( ) ( ) ( )[ ]321: ∈= Ttxtxtxtx

( ) ℜ∈ty is the system output, initial value, 1f∆ is uncertain function, indicate the parameters of the system

0≠b . Besides, in order to ensure the existence and uniqueness of the solution, we assume that

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Class of Chaotic Systems

Shou University, Kaohsiung, Taiwan

equalities, a new state uncertain nonlinear systems

ensure the global exponential stability of the resulting error system. In addition, the guaranteed exponential convergence rate can be precisely calculated. Finally, some numerical simulations will be given to demonstrate the

ult.

This paper is organized as follows. The problem are presented in Section

simulations are given in Section to illustrate the main result. Finally, conclusion

remarks are drawn in Section 4. Throughout this dimensional real space,

the Euclidean norm of the column

he absolute value of a real

ROBLEM FORMULATION AND MAIN

the following uncertain

(1a) (1b) (1c) (1d)

]Tx30 , (1e)

3ℜ∈ is the state vector,

is the system output, [ ]Txxx 302010 is the is uncertain function, and ℜ∈ba,

parameters of the systems, with 0<a and n order to ensure the existence and

uniqueness of the solution, we assume that all the

International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456


functions of 1f∆ and ( ) { }4,3,2, ∈∀⋅ ifi are smooththe inverse function of 4f exists. The chaotic sprott E system is a special case of systems (1) with

212 xf = , 13 =f , 14 4xf −= , and 1−=a . It is a well

fact that since states are not always available for direct measurement, particularly in the event of sensor failures, states must be estimated. The paper is to search a novel state observer uncertain nonlinear systems (1) such that exponential stability of the resulting error systems can be guaranteed. Before presenting the main result, the state reconstructibility is provided as follows. Definition 1 The uncertain nonlinear systems (1) are state reconstructible if there exist a state observer

( ) 0,, =yzzf & and positive numbers κ and ( ) ( ) ( ) ( ) 0,exp: ≥∀−≤−= tttztxte ακ ,

where ( )tz represents the reconstructed state of systems (1). In this case, the positive number called the exponential convergence rate. Now, we are in a position to present the main results for the exponential state observer of uncertain (1). Theorem 1. The uncertain systems (1) are exponentially state reconstructible. Moreover, a suitable state observer is given by

( ) ( ) ( ) ,11

31

41

−= − tyb

ftyb

ftz & (2a)

( ) ( ) ( ) ( )( ),, 31222 tztzftaztz +=& (2b)

( ) ( ) 0,1

3 ≥∀= ttyb

tz . (2c)

In this case, the guaranteed exponentiarate is given by a−=:α . Proof. For brevity, let us define the observer error

( ) ( ) ( ) { }3,2,1,: ∈∀−= itztxte iii and 0≥t . (3) From (1d) and (2c), one has

( ) ( ) ( )

( ) ( )tyb

tyb

tztxte

11333

−=

−=

.0,0 ≥∀= t (4)



are smooth and chaotic sprott E

(1) with 321 xxf =∆ ,

. It is a well-known fact that since states are not always available for direct

vent of sensor he aim of this

state observer for the (1) such that the global

the resulting error systems can

nting the main result, the state reconstructibility is provided as follows.

are exponentially state reconstructible if there exist a state observer

and α such that

the reconstructed state of (1). In this case, the positive number α is

called the exponential convergence rate.

a position to present the main results uncertain systems

exponentially state , a suitable state observer is

the guaranteed exponential convergence

observer error

From (1c), it can be readily obtained that( )( ) ( ) ( )( )txftxtxf 33314 −= & .

It results that

( ) ( ) ( )( )( )

( ) ( )

−=

−=

−

−

tyb

ftyb

f

txftxftx

113

14

3331

41

&

&

Thus, one has

( ) ( ) ( )

( ) ( )

( ) ( )

−−

−=

−=

−

−

tyb

ftyb

f

tyb

ftyb

f

tztxte

11

11

31

4

31

4

111

&

&

,0,0 ≥∀= t in view of (5) and (2a). In addition(4), and (6), it is easy to see that

( ) ( ) ( )( ) ( ) ( )( )[ ]( ) ( ) ( )( )[ ]

( ) ( ) ( )( )[ ]( ) ( ) ( )( )[ ]

( ) ( )[ ]tztxa

txtxftaz

txtxftax

tztzftaz

txtxftax

tztxte

22

3122

3122

3122

3122

222

,

,

,

,

−=+−

+=+−

+=−= &&&

( ) .0,2 ≥∀= ttae It follows that

( ) ( ) .0,022 ≥∀=− ttaete& Multiplying with ( )at−exp yields

( ) ( ) ( ) ( ) 0expexp 22 =−⋅−−⋅ attaeatte&

Hence, it can be readily obtained tha

( ) ( )[ ].0,0

exp2 ≥∀=−⋅t

dt

atted

Integrating the bounds from 0

( ) ( )[ ],00

exp

00

2 ==−⋅∫∫ dtdt

dt

atted tt

This implies that ( ) ( ) ( ) 0,exp022 ≥∀⋅= tatete .

Thus, from (4), (6), and (7), we have

( ) ( ) ( ) ( )( ) ( ) .0,exp02

23

22

21

≥∀⋅=

++=

tate

tetetete

Consquently, we conclude that tsuitable state observer with the guaranted exponential convergence rate a−=α . This completes the proof.

International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456-6470

Dec 2018 Page: 1159

From (1c), it can be readily obtained that

(5)

(6)

In addition, from (1b), (1c), is easy to see that

yields .0,0 ≥∀ t

Hence, it can be readily obtained that

and t , it results

0≥∀ t .

(7)

, we have

conclude that the system (2) is a the guaranted exponential

. This completes the proof. □



3. NUMERICAL SIMULATIONS Consider the uncertain nonlinear system

321 xxcf ⋅∆=∆ , 212 xf = , 13 =f , (8a)

14 4xf −= , 1−=a , 2=b , 11 ≤∆≤− c . (8b) By Theorem 1, we conclude that the systems (1) with (8) is exponentially state reconstructible by the state observer

( ) ( ) ,4

1

8

11 +−= tytz & (9a)

( ) ( ) ( ),2122 tztztz +−=& (9b)

( ) ( ) .0,2

13 ≥∀= ttytz (9c)

The typical state trajectory of the uncertain (1) with (8) is depicted in Figure 1. Furthermoretime response of error states is depicted in FigFrom the foregoing simulations results, it is seen that the uncertain systems (1) with (8) are state reconstructible by the state observer othe guaranted exponential convergence rate 4. CONCLUSION In this paper, a class of uncertain chaotic and nonchaotic systems has been introduced observation problem of such systemstudied. Based on the time-domain approach with integral and differential equalities, a observer for a class of uncertain nonlinear been constructed to ensure the global exponential stability of the resulting error system. guaranteed exponential convergence rate precisely calculated. Finally, numerical simulations have been presented to exhibit the effectiveness feasibility of the obtained results. ACKNOWLEDGEMENT The author thanks the Ministry of Science and Technology of Republic of China for supporting this work under grants MOST 106-2221MOST 106-2813-C-214-025-E, and MOST E-214-030. Besides, the author is grateful to Professor Jer-Guang Hsieh for the useful REFERENCES 1. S. Xiao and Y. Zhao, “A large class of chaotic

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, we conclude that the uncertain is exponentially state

uncertain systems Furthermore, the

is depicted in Figure 2. From the foregoing simulations results, it is seen that

are exponentially state reconstructible by the state observer of (9), with the guaranted exponential convergence rate 1=α .

chaotic and non- and the state

observation problem of such systems has been domain approach with

, a novel state uncertain nonlinear systems has

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Ministry of Science and of Republic of China for supporting this

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Figure 1: Typical state trajectory

Figure

0-8

-6

-4

-2

0

2

4

6

8

x1(t

); x

2(t);

x3(t)

0-0.5

0

0.5

1

1.5

2

e1(t

); e

2(t)

; e3

(t)

e1=e2=0

e3



ing Takagi-Sugeno,” 44, pp. 114-122,

14. A. Sassi, H.S. Ali, M. Zasadzinski, and K. Abderrahim, “Adaptive observer design for a class of descriptor nonlinear systems,” European Journal of Control, vol. 44, pp. 90

trajectory es of the uncertain nonlinear systems (1)

ure 2: The time response of error states.

50 100 150t (sec)

x1: the Blue Curve

x2: the Green Curve

x3: the Red Curve

5 10 15t (sec)


Dec 2018 Page: 1161

A. Sassi, H.S. Ali, M. Zasadzinski, and K. Abderrahim, “Adaptive observer design for a class of descriptor nonlinear systems,” European Journal of Control, vol. 44, pp. 90-102, 2018.

(1) with (8).

Exponential State Observer Design for a Class of Uncertain Chaotic and Non-Chaotic Systems

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